Asymptotic Curvature of Moduli Spaces for Calabi–Yau Threefolds

Motivated by the classical statements of Mirror Symmetry, we study certain Kähler metrics on the complexified Kähler cone of a Calabi–Yau threefold, conjecturally corresponding to approximations to the Weil–Petersson metric near large complex structure limit for the mirror. In particular, the naturally defined Riemannian metric (defined via cup-product) on a level set of the Kähler cone is seen to be analogous to a slice of the Weil–Petersson metric near large complex structure limit. This enables us to give counterexamples to a conjecture of Ooguri and Vafa that the Weil–Petersson metric has non-positive scalar curvature in some neighborhood of the large complex structure limit point.